A300820 Length of the longest sequence of consecutive primes in the prime factorization of n. a(1) = 0.

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3

Offset: 1

Author

Antti Karttunen, Mar 21 2018

Keywords

• nonn

Examples

			For n = 350 = 2 * 5^2 * 7 = prime(1) * prime(3)^2 * prime(4), the longest stretch of consecutive primes is from prime(3) to prime(4), with length 2, thus a(350) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A038374, A087207, A104210 (positions of terms > 1), A296210.

Cf. also A167447.

Programs

PARI

A300820(n) = if(omega(n)<=1, omega(n), my(pis=apply(p->primepi(p),factor(n)[,1]),el=1,m=1); for(i=2,#pis,if(pis[i] == (1+pis[i-1]),el++; m = max(m,el), el=1)); (m));

PARI

a(n) = {if(n == 1, return(0)); my(res = 1, f = factor(n)[, 1]~, t = 1);
for(i = 1, #f - 1, if(f[i+1]==nextprime(f[i]+1), t++, res = max(res, t);  t = 1)); max(res, t)} \\ David A. Corneth, Mar 21 2018

Formula

For n > 1, a(n) = A038374(A087207(n)).

For n >= 0, a(A002110(n)) = n. [Primorials give the positions of the records = the first occurrence of each n.]